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Find the partial derivatives with respect to (a) $x$ and (b) $y$.$$f(x, y)=2 x^{2}+3 y^{2}+4 x^{3} y^{4}+9 x-2 y+7$$

$4 x+12 x^{2} y^{4}+9$(b) $6 y+16 x^{3} y^{3}-2$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Find the partial derivativ…

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Compute the given partial …

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Find both first partial de…

So if we want to find the partial derivative of this with respect to X, and why so memorable we're going to do is, um so first for part A. If we want to take the partial of this with respect to X this Dell by Dell X Here we treat all of these wise here as if they are constants. So just like how we would treat seven or nine or two is a constant. We're going to treat all of these wise as a constant right now. So when we go ahead to do that so we would normally write this as f partial X. So, like this subscript of it ex saying That's the partial derivative who took. And then So we take the partial of, or just pretty much like taking the derivative of two x squared. That would be power rules. So it's gonna be four x now three y squared. Why is a constant? So if we square constant, that's still a constant. If we multiply constant, constant, so constant. So this overall, at least with respect to X, is a constant, so that's just going to become zero. Now we have four execute y to the fourth. Now four y to the fourth is a constant. Remember, we could just kind of factor that out when we're taking the derivative. And then on Lee focus on X cute. So the derivative X cubed is going to be three x squared. And then so Nynex just like we would take derivatives. So just nine now to why again? Why is a constant We're assuming times too, So that would be just zero. And then seven is also just a constant. So we could go ahead and, um, get rid of all those zeros. So we have four x plus Multiply the four in the three. So b 12, um, I will just write the X in front X squared. Why? To the fourth. So this is our partial with respect to X. Now, let me go ahead and pick this up and scoot this down to get the partial with respect to why we're going to do exactly what we just did. But this time we do del by Dell, Why? And so on the left that's going to give us f partial with respect to y. So we say f sub y like that. And now we're going to assume all of these X is here are constants. So when we come over here to take the derivative two x squared just like when we took the river of three Wise, where this just a constant So that would be zero plus not three y squared. We would just use power rules that would be three times to so six. Why? Plus now X cubed is a constant with respect to why so we just have four x cubed out front, and then we take the derivative of y to the fourth, which would be for why cute nine times X X is a constant times, another constant still a constant. So it'll be plus zero negative too wide. I would just leave us with negative two and then derivative of seven would just be zero, and then we can go ahead and clean this up a little bit, so that tells us our personal with respect to Why is going to be so six y plus 16 x cubed by cubed minus two. And so then this is our partial with respect to y. So again, just remember, whenever we take thes partial derivatives, whichever partial we're taking, we just assume all other variables we have are a constant

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